An adaptive finite element method for high-frequency scattering problems with smoothly varying coefficients

TL;DR

This paper presents a novel adaptive finite element method for high-frequency acoustic scattering problems with smoothly varying coefficients, leveraging a time-domain approach and mesh adaptation to improve efficiency and accuracy.

Contribution

The paper introduces a new time-domain based adaptive finite element method with front-tracking mesh adaptation for high-frequency scattering, reducing degrees of freedom needed for accuracy.

Findings

Efficient handling of high-frequency problems with reduced degrees of freedom.

Effective reconstruction of time-harmonic solutions from time-domain data.

Applicability to problems with external sources like point sources and scatterers.

Abstract

We introduce a new numerical method for solving time-harmonic acoustic scattering problems. The main focus is on plane waves scattered by smoothly varying material inhomogeneities. The proposed method works for any frequency $\omega$, but is especially efficient for high-frequency problems. It is based on a time-domain approach and consists of three steps: \emph{i)} computation of a suitable incoming plane wavelet with compact support in the propagation direction; \emph{ii)} solving a scattering problem in the time domain for the incoming plane wavelet; \emph{iii)} reconstruction of the time-harmonic solution from the time-domain solution via a Fourier transform in time. An essential ingredient of the new method is a front-tracking mesh adaptation algorithm for solving the problem in \emph{ii)}. By exploiting the limited support of the wave front, this allows us to make the number of the required degrees of freedom to reach a given accuracy significantly less dependent on the frequency $\omega$. We also present a new algorithm for computing the Fourier transform in \emph{iii)} that exploits the reduced number of degrees of freedom corresponding to the adapted meshes. Numerical examples demonstrate the advantages of the proposed method and the fact that the method can also be applied with external source terms such as point sources and sound-soft scatterers. The gained efficiency, however, is limited in the presence of trapping modes.